Transcript AOSS 401, Fall 2007 Lecture 20 October 29, 2007

AOSS 401, Fall 2007
Lecture 24
November 07, 2007
Richard B. Rood (Room 2525, SRB)
[email protected]
734-647-3530
Derek Posselt (Room 2517D, SRB)
[email protected]
734-936-0502
Class News
November 07, 2007
• Homework 6 (Posted this evening)
– Due Next Monday
• Important Dates:
– November 16: Next Exam (Review on 14th)
– November 21: No Class
– December 10: Final Exam
Weather
• National Weather Service
– http://www.nws.noaa.gov/
– Model forecasts:
http://www.hpc.ncep.noaa.gov/basicwx/day07loop.html
• Weather Underground
– http://www.wunderground.com/cgibin/findweather/getForecast?query=ann+arbor
– Model forecasts:
http://www.wunderground.com/modelmaps/maps.asp
?model=NAM&domain=US
Rest of Course
• Wrap up quasi-geostrophic theory (Chapter 6)
– Potential vorticity
– Vertical velocity
– Will NOT do Q vectors
• We will have a lecture on the Eckman layer (Chapter 5)
– Boundary layer, mix friction with rotation
• We will have a lecture on Kelvin waves (Chapter 11)
– A long wave in the tropics
• There will be a joint lecture with 451 on hurricanes
(Chapter 11)
• Computer homework (perhaps lecture) on modeling
• Special topics?
Material from Chapter 6
• Quasi-geostrophic theory
• Quasi-geostrophic vorticity
– Relation between vorticity and geopotential
• Geopotential prognostic equation
• Quasi-geostrophic potential vorticity
Scaled equations in pressure coordinates
(The quasi-geostrophic (QG) equations)
Dg v g
  f 0k  v a   y k  v g
momentum equation
Dt
1
v g  k  
f0
ua v a 


0
x
y p
 
  
J
  v g      
t
 p 
p
geostrophic wind
continuity equation
thermodynamic
equation
R
R d To d ln 0 
with  
and stability parameter
=
cp
p
dp
Approximations in the
quasi-geostrophic (QG) theory
v  vg  va
with v a v g  O(Ro)  0.1
Dg 
Dv Dg v g



wit h
  ug  v g
Dt
Dt
Dt t
x
y
1
v g  k  
f0
Midlatitude - plane approximation
:
f 
2cos( 0 )
f  f 0    y  f 0  y = f 0 
y
a
y  0
with y = a( -  0 )
(a= radius of t he earth)
constant f 0  2sin( 0 )
Ttot (x, y, p,t)  T0 ( p)  T(x, y, p,t)
with
T0
T

p
p
Quasi-geostrophic equations cast in terms of geopotential
and omega.
 f 0  
 f0


 J
(
)   ( Vg  
)  f0
 f0
( )
p  p t
p 
p
p
p p
1 2 

1 2

 f0
 Vg  (    f )
f0
t
p
f0
THERMODYNAMIC EQUATION
VORTICITY EQUATION
Add these equations to eliminate omega and we have a
partial differential equation for geopotential tendency
(assume J=0)
 f 0  
 f


 J
(
)   ( 0 Vg  
)  f0
 f0
( )
p  p t
p 
p
p
p p
1 2 

1

 f0
 Vg  (  2   f )
f0
t
p
f0
f 02
 f 02  
1 2


(  (
))
  f 0 Vg  (    f )  (
Vg  (
))
p  p t
f0
p

p
2
GEOPOTENTIAL TENDENCY EQUATION
Add these equations to eliminate omega and we have a
partial differential equation for geopotential tendency
(assume J=0)
f 02
 f 02  
1 2


(  (
))
  f 0 Vg  (    f )  (
Vg  (
))
p  p t
f0
p 
p
2
f0 * Vorticity Advection
Add these equations to eliminate omega and we have a
partial differential equation for geopotential tendency
(assume J=0)
f 02
 f 02  
1 2


(  (
))
  f 0 Vg  (    f )  (
Vg  (
))
p  p t
f0
p 
p
2
Thickness Advection
Advection of vorticity
ζ < 0; anticyclonic
 Advection of ζ tries to propagate the wave this way 
ΔΦ > 0
٠
Φ0 - ΔΦ
B
L
Φ0
L
H
 Advection of f tries to propagate the wave this way 
٠
Φ0 + ΔΦ
٠
y, north
A
x, east
ζ > 0; cyclonic
C
ζ > 0; cyclonic
Relationship between
upper troposphere and surface
vorticity advection
thickness advection
To think about this
• Read and re-read pages 174-176 in the
text.
Idealized vertical cross section
Real baroclinic disturbances
Great web page with current maps:
http://www.meteoblue.ch/More-Maps.79+M5fcef4ad590.0.html
Personalize your maps (create a
login):
http://my.meteoblue.com
Real baroclinic disturbances:
850 hPa temperature and geopot. thickness
cold air
advection,
enhances
trough
warm air
advection
east of the
surface
low,
enhances
the ridge
Real baroclinic disturbances:
500 hPa rel. vorticity and mean SLP
sea level
pressure
Positive
vorticity,
pos. vorticity
advection,
increase in
cyclonic
vorticity
Real baroclinic disturbances:
500 hPa geopot. height and mean SLP
Upper level
systems
lags behind
(to the west):
system still
develops
With the benefit of hindsight and
foresight let’s look back.
QG vorticity equation
 g

 f0
 v g  ( g  f )
t
p
 g

 f0
 v g   g  v g
t
p
THINKING
ABOUT
THESE
TERMS
Advection of
relative vorticity

Advection of
planetary
vorticity
Stretching
term
Competing
QG vorticity equation
 g

 f0
 v g  ( g  f )
t
p
 g

 f0
 v g   g  v g
t
p
Advection of
relative vorticity
WHAT
ABOUT THIS
TERM?

Stretching
term
Advection of
planetary
vorticity
Competing
Consider our simple form of potential
vorticity
From scaled equation, with assumption of constant
density and temperature.
Dhorizontal   f
(
)0
Dt
H
f
 potent ialvort icit y
H
There was the assumption that the layer of fluid
was shallow.
Fluid of changing depth
What if we have something like this, but the fluid is
an ideal gas?
Add these equations to eliminate omega and we have a
partial differential equation for geopotential tendency
(assume J=0)
f 02
 f 02  
1 2


(  (
))
  f 0 Vg  (    f )  (
Vg  (
))
p  p t
f0
p 
p
2
Still looks a lot like time rate of
change of vorticity
Quasi-Geostrophic
potential vorticity (PV) equation
Simplify the last term of the geopotential tendency
equation by applying the chain rule:
  = 0 Why?
  f 02   f 02 v g
v g  
  


p   p   p
p 

stability parameter
=
R d To dln 0 
p
dp
Quasi-Geostrophic
potential vorticity (PV) equation
Simplify the last term of the geopotential tendency
equation by applying the chain rule:
  = 0 Why?
  f 02   f 02 v g
v g  
  


p   p   p
p 
THERMAL WIND RELATION

v g

  

f0
 k  
p
 p 
stability parameter
=
R d To dln 0 
p
dp
Quasi-Geostrophic
potential vorticity (PV) equation
Simplify the last term of the geopotential tendency
equation by applying the chain rule:
  = 0 Why?
  f 02   f 02 v g
v g  
  


p   p   p
p 
Leads to the
conservation law:
stability parameter
=
R d To dln 0 
p
dp
Dg
q
 v g  q
q0
t
Dt
Quasi-geostrophic

potential vorticity:
 1 2
 f 0 
q      f  

p  p 
 f 0
Conserved following the geostrophic motion
Imagine at the point flow decomposed
into two “components”
A “component” that flows around the point.
Vorticity
• Related to shear of the velocity field.
∂v/∂x-∂u/∂y
Imagine at the point flow decomposed
into two “components”
A “component” that flows into or away from the point.
Divergence
• Related to stretching of the velocity field.
∂u/∂x+∂v/∂y
Potential vorticity (PV): Comparison
Barotropic PV:
PV 
Units:
g  f
m-1s-1
h
 1 2
Quasi f 0 
q      f  

geostrophic PV:
p  p 
 f 0


Ertel’s PV: PV  (  f )(g
)
p

s-1
K kg-1 m2 s-1
THESE ARE LIKE STRETCHING IN THE VERTICAL

QG vorticity equation
 g

 f0
 v g  ( g  f )
t
p
 g

 f0
 v g   g  v g
t
p
Advection of
relative vorticity
WHAT
ABOUT THIS
TERM?

Stretching
term
Advection of
planetary
vorticity
Competing
Fluid of changing depth
What if we have something like this, but the fluid is
an ideal gas?
Conversion of thermodynamic energy to vorticity,
kinetic energy. Again the link between the thermal
field and the motion field.
Two important definitions
• barotropic – density depends only on
pressure. And by the ideal gas equation,
surfaces of constant pressure, are
surfaces of constant density, are surfaces
of constant temperature (idealized
assumption).
= (p)
• baroclinic – density depends on pressure
and temperature (as in the real world).
= (p,T)
Barotropic/baroclinic atmosphere
Barotropic:
p
p + p
p + 2p
T
T+T
T+2T
T
Baroclinic:
T+T T+2T
p
p + p
p + 2p
ENERGY IN HERE THAT IS CONVERTED TO MOTION
Barotropic/baroclinic atmosphere
Barotropic:
p
p + p
p + 2p
T
T+T
T+2T
T
Baroclinic:
T+T T+2T
p
p + p
p + 2p
DIABATIC HEATING KEEPS BUILDING THIS UP
• NOW WOULD BE A GOOD TIME FOR A
SILLY STORY
VERTICAL VELOCITY
Vertical motions: The relationship
between w and 
Dp p
p

  v h  p  w
Dt t
z
p
   v g  p  v a  p  wg
t
=0
hydrostatic equation
p
   v a  p  wg
t
≈ 10 hPa/d
≈ 1m/s 1Pa/km
≈ 1 hPa/d
≈ 100 hPa/d
with the help of scale analysis (free troposphere)
  wg
Link between  and the ageostrophic wind

0
p

 p  (v g  v a ) 
0
p
ug v g 





(v
)

0


p
a
y p
p
 x
 p  vh 
 
1 
 1  

(
)

(
)



(v
)

0


p
a
y f x p
p
x f y
assume f is approximat ely const ant

  p  (v a )
p

if v h  v g 
0
p
=0
Links the horizontal and vertical motions. Since
geostrophy is such a good balance, the vertical motion is
linkedto the divergence of the ageostrophic wind (small).
Vertical pressure velocity 
For synoptic-scale (large-scale) motions in midlatitudes
the horizontal velocity is nearly in geostrophic balance.
Recall: the geostrophic wind is nondivergent (for
constant Coriolis parameter), that is
ug v g
  vg 

0
x y
Horizontal divergence is mainly due to small departures

from geostrophic
balance (ageostrophic wind).
Therefore: small errors in evaluating the winds <u> and <v>
 u  v 
 ( p1)   ( p2 )  ( p1  p2 )



x

y

p
lead to large errors in . The kinematic method is inaccurate.
Think about this ...
• If I have errors in data, noise.
• What happens if you average that data?
• What happens if you take an integral over
the data?
• What happens if you take derivatives of
the data?
Estimating the vertical velocity:
Adiabatic Method
Start from thermodynamic equation in p-coordinates:
T
T
T
J
u
v
 S p 
t
x
y
cp
Assume that the diabatic heating term J is small (J=0),
re-arrange the equation

T
T
T 
  S   u  v 
x
y 
t
1
p

Sp:Stability
parameter
- (Horizontal temperature
advection term)
Estimating the vertical velocity:
Adiabatic Method
T
If local time tendency  0 (steady state)
t
 T
T  v h  T
  S u  v  
y 
Sp
 x
1
p
Horizontal temperature
advection term
Stability parameter
If T/t = 0 (steady state), J=0 (adiabatic) and Sp > 0 (stable):
 • then warm air advection:
 < 0, w ≈ -/g > 0 (ascending air)
• then cold air advection:
 > 0, w ≈ -/g < 0 (descending air)
Adiabatic Method
• Based on temperature advection, which is
dominated by the geostrophic wind, which
is large. Hence this is a reasonable way
to estimate local vertical velocity when
advection is strong.
Estimating the vertical velocity:
Diabatic Method
Start from thermodynamic equation in p-coordinates:
T
T
T
J
u
v
 S p 
t
x
y
cp
If you take an average over space and time, then the
advection and time derivatives tend to cancel out.

J 
   S  
 cp 
1
p
Diabatic term
mean meridional circulation
Conceptual/Heuristic Model
•Observed characteristic
behavior
•Theoretical constructs
•“Conservation”
•Spatial Average or
Scaling
•Temporal Average or
Scaling
Yields
Relationship between
parameters if observations and
theory are correct
Plumb, R. A. J. Meteor. Soc. Japan, 80, 2002
One more way for vertical velocity
Quasi-geostrophic equations cast in terms of geopotential
and omega.
 f 0  
 f0


 J
(
)   ( Vg  
)  f0
 f0
( )
p  p t
p 
p
p
p p
1 2 

1 2

 f0
 Vg  (    f )
f0
t
p
f0
THERMODYNAMIC EQUATION
VORTICITY EQUATION
ELIMINATE THE GEOPOTENTIAL AND GET AN EQUATION FOR OMEGA
Quasi-Geostrophic Omega Equation
1.) Apply the horizontal Laplacian operator to the
QG thermodynamic equation
2.) Differentiate the geopotential height tendency
equation with respect to p
3.) Combine 1) and 2) and employ the chain rule
of differentiation (chapter 6.4.1 in Holton, note
factor ‘2’ is missing in Holton Eq. (6.36), typo)
 1 2
 2 f 02 2  2 f 0  v g

    

 
2 
  p
  p

 f 0


f 

Advection of absolute vorticity
by the thermal wind
Vertical Velocity Summary
• Though small, vertical velocity is in some ways
the key to weather and climate. It’s important to
waves growing and decaying. It is how far away
from “balance” the atmosphere is.
• It is astoundingly difficult to calculate. If you use
all of these methods, they should be equal. But
using observations, they are NOT!
• In fact, if you are not careful, you will not even to
get them to balance in models, because of
errors in the numerical approximation.
One more summary of the midlatitude wave
Idealized (QG) evolution of a baroclinic disturbance
(Read and re-read pages 174-176 in the text.)
500 hPa
geopotential
p at the
surface
H
+ pos. vorticity
advection
- cold air
advection
L
+ warm air
advection
- neg. vorticity
advection
p at the
surface
Waves
• The equations of motion contain many forms
of wave-like solutions, true for the atmosphere
and ocean
• Some are of interest depending on the
problem: Rossby waves, internal gravity
(buoyancy) waves, inertial waves,
inertial-gravity waves, topographic waves,
shallow water gravity waves
• Some are not of interest to meteorologists,
e.g. sound waves
• Waves transport energy, mix the air
(especially when breaking)
Waves
• Large-scale mid-latitude waves, are critical for
weather forecasting and transport.
• Large-scale waves in the tropics (Kelvin waves,
mixed Rossby-gravity waves) are also important,
but of very different character.
• This is true for both ocean and atmosphere.
• Waves can be unstable. That is they start to
grow, rather than just bounce back and forth.
• And, with that, Chapter 6, of Jim Holton’s
book rested comfortably in the mind of the
students.
Below
• Basic Background Material
Couple of Links you should know
about
• http://www.lib.umich.edu/ejournals/
– Library electronic journals
• http://portal.isiknowledge.com/portal.cgi?In
it=Yes&SID=4Ajed7dbJbeGB3KcpBh
– Web o’ Science
A nice schematic
• http://atschool.eduweb.co.uk/kingworc/dep
artments/geography/nottingham/atmosphe
re/pages/depressionsalevel.html
Mid-latitude cyclones:
Norwegian Cyclone Model
• http://www.srh.weather.gov/jetstream/syno
ptic/cyclone.htm
Tangential coordinate system
R=acos()
Place a coordinate
system on the surface.
Ω
R a
Φ
Earth
x = east – west
(longitude)
y = north – south
(latitude)
z = local vertical
or
p = local vertical
Tangential coordinate system
f=2Ωsin()
Relation between
latitude, longitude and x
and y
Ω
=2Ωcos()/a
R a
Φ
Earth
dx = acos() dl
lis longitude
dy = ad
 is latitude
dz = dr
r is distance from center
of a “spherical earth”
Equations of motion in pressure coordinates
(using Holton’s notation)
DV
 fk  V  
Dt
u v


(  )p 
 V 
0
x y
p
p
T
T
T
T
J
u
v
 S p 
 V  T  S p 
t
x
y
t
cp

RT
 ln 
   
; S p  T
p
p
p
V  ui  vj  horizontalvelocity ;   pot entialtemperatu
re
D( ) 

Dp
 ) p  (V  ) p  
;
Dt t
p
Dt
timeand horizontalderivatives at constantpressure
(oftennot explicitlywritten)
Scale factors for “large-scale” mid-latitude
U  10 m s
-1
P  10 hP a
W  1 cm s units!
  1 kg m
L  10 m
 /   10 2
-1
6
H  10 m
-3
4
L / U  10 s
5
f 0  10-4 s 1
f

 10-11 m -1s -1
y
Scaled equations of motion in pressure coordinates
Vg 
1
k  
f0
Definition of
geostrophic wind
Dg Vg
  f 0k  Va  yk  Vg
Momentum
equation
Dt
ua va 


0
x
y p
Continuity
equation
J
R

 

V






;




g
p
cp
 t
 p
Thermodynamic
Energy equation
Weather
• National Weather Service
– http://www.nws.noaa.gov/
– Model forecasts:
http://www.hpc.ncep.noaa.gov/basicwx/day07loop.html
• Weather Underground
– http://www.wunderground.com/cgibin/findweather/getForecast?query=ann+arbor
– Model forecasts:
http://www.wunderground.com/modelmaps/maps.asp
?model=NAM&domain=US